Authors: Russell Letkeman
The sequence of sets of $Z_n$ where n is a primorial gives us a surprisingly simple and elegant tool to investigate many properties of the prime numbers and their distributions through analysis of their gaps. A sensible reason to study multiplication on these boundaries is a natural construction exists which evolves the sets from one primorial boundary to the next via the sieve of Eratosthenes. To this we add a parallel study of gaps of various lengths all of which together informs what we call the S model. These sets can be individually broken down into their respective p.m.f. from which we build a purely probabilistic model of gap distributions. We use this framework to prove some interesting distributional properties of the prime numbers.
Comments: 25 Pages. corrected
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